Electrochemical Modeling, Supervision and Control of Lithium-Ion Batteries

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Electrochemical Modeling, Supervision and Control of

Lithium-Ion Batteries

Thesis submitted by Luis D. COUTO

in fulfilment of the requirements of the PhD Degree in Engineering Sciences

and Technology (“Docteur en Sciences de l’Ingénieur et Technologie”)

Academic year 2018-2019

Supervisor: Professor Michel KINNAERT

Department of Control Engineering and System Analysis

Thesis jury:

Philippe BOGAERTS (Université libre de Bruxelles, Chair)

Johan GYSELINCK (Université libre de Bruxelles, Secretary)

Emanuele GARONE (Université libre de Bruxelles)

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“It is interesting to note that most of the early contributors to estimation theory were primarily astronomers rather than mathematicians. They used mathematics as a means to an end. Then, as now, the most outstanding and lasting contributions to theory were driven by practical engineering interests. “There is nothing so practical as a good theory”. ”

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UNIVERSITÉ LIBRE DE BRUXELLES

Abstract

École Polytechnique de Bruxelles

Department of Control Engineering and System Analysis Submitted for the PhD Degree in Engineering Sciences and Technology

Electrochemical Modeling, Supervision and Control of Lithium-Ion Batteries

by Luis D. COUTO

This thesis develops an advanced battery monitoring and control system based on the electrochemical principles that govern lithium-ion battery dynamics. This work is motivated by the need of having safer and better energy storage systems for all kind of applications, from small scale portable electronics to large scale renewable energy storage. In this context, lithium-ion batteries have become the enabling technology for energy autonomy in appliances (e.g. mobile phone, electric vehicle) and energy self-consumption in households. However, batteries are oversized and pricey, might be unsafe, are slow to charge and may not equalize the lifetime of the application they are intended to power. This work tackles these different issues.

This document first introduces the general context of the battery management problem, as well as the particular issues that arise when modeling, supervising and controlling the battery short-term and long-term operation. Different solutions coming from the literature are reviewed, and several standard tools borrowed from control theory are exposed. Then, starting by well-known contributions in electro-chemical modeling, we proceed to develop reduced-order models for the battery operation including degradation mechanisms, that are highly descriptive of the real phenomena taking place. This modeling framework is the cornerstone of all the monitoring and control development that follows.

Next, we derive a battery diagnosis system with a twofold objective. First, indi-cators for internal faults affecting the battery state-of-health are obtained. Secondly, detection and isolation of sensor faults is achieved. Both tasks rely on state observers designed from electrochemical models to perform state estimation and residual gen-eration. Whereas the former solution resorts to system identification techniques for health monitoring, the latter solution exploits fault diagnosis for instrumentation assessment.

We then develop a feedback battery charge strategy able to push in performance while accounting for constraints associated to battery degradation. The fast and safe charging capabilities of the proposed approach are ultimately validated through long-term cycling experiments. This approach outperforms widely used commercial charging strategies in terms of both charging speed and degradation.

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Acknowledgements

This dissertation is the result of four years of research at the Université Libre de Bruxelles, Department of Control Engineering and System Analysis. It is, however, the combined effort of several high-quality researchers and extraordinary people who have accompanied me during this journey. In this page, I will try to acknowledge all these people, and I apologize in advance if you, the reader, should have been mentioned.

First and foremost, I want to thank my advisor, Prof. M. Kinnaert, to whom I dedicate this thesis. I recall when he interviewed me four years ago, and how he trusted me and gave me the opportunity to get on this boat. He is one of the best researchers I have met (possibly I will ever meet), and he is a better person than a researcher. He taught me how to do research with an open mind and a critical spirit. Many thanks for your advice, patience and kindness, Michel. I will always be grateful.

I would also like to thank Prof. E. Garone. Although he was not directly my advisor, I felt like he was. He was always there every time I needed him. He is one of the smartest people I know, and we have been validating intuitions that he had four years ago. There are still some thoughts that we need to test, but he is probably right. He always is. Continuing with professors at the ULB, I want to thank the members of my thesis committee, namely Prof. P. Bogaerts and Prof. J. Gyselinck. Their curiosity and thorough feedback during the committee meetings will always be appreciated. Finally, I would also like to thank Prof. A. Léonard, with whom I collaborated for my first journal paper as a first author. He provided invaluable help in the achievement of the referred paper.

My gratitude also goes to the people at the Department that helped me during my first steps here. Without any priority order, I would like to thank Marco, who was one of my closest friends at the beginning of this adventure. Special thanks for Jingjing, who was always there to support me during the hardest initial moments of hesitation and uncertainty. Also thanks to Julien, my first office mate and dear friend. He had the patience to guide me and he believed in me from the start. He still does it. Also thanks to Laurent, Pascale, Serge and Prof. R. Hanus, who received me with open arms when I arrived. I want to give a shout-out to my other friends in the lab, namely Tam, Raffaele, Johnny, Sandra, Chris L., Robin, Chris M., Silvia, Nico, Sadja, Medhi, Silvane, Andres, Alejandro, Alberto, Satoshi, Kelly and Bryan. Most of them have tolerated me during the last months of the thesis, and that requires special abilities.

I also want to give a warm thanks to Prof. S. Moura from UC Berkeley. He received me in his department for seven months and without his collaboration, the results of this work would not have been possible. Besides letting us use his facilities, he was always open to have deep discussions and allowed me to contribute in different projects. Thank you very much Prof. Moura for giving me the opportunity to live such an intellectually stimulating experience. He leads an awesome team of people. Among them, I would like to thank Dong, Saehong, Zach and Hector, who made me feel as a part of the group from the beginning.

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S. Di Scipio who taught me separation processes and always was a close friend. They both are from the USB.

Last but not least, I want to thank my friends from my home country university and school, as well as my friends from life and family. I remember fondly Edu, Eli, Mou, Kathy, Dani, Fabi, Gian, Amaia, Kevin, Valen, Leo, Sergio and Dario, from the years back at the university. Special mention for Luis F., Santi, David, Pedro, Popi, Eu, Ori and Carito, my unforgettable and invaluable friends from school. They showed me what the true friendship is when you are a kid and you simply do not care about e.g. appearance or money. Thanks to my friends of life, who I met just because we happened to be in the right place at the right time, like Sylvie, Lau, Loui, Anita, Sirine, Emily, Ionel, Clau, Isa, Rebecca, Sophie, Alejandra and Ruta. Many thanks Loui, for holding me there the last 100 m of the race. And the most important thanks for the end, to my family. My mom, dad and sister, together with my aunts, cousins, grandparents and godfathers. I could not be here if it were not for you.

Finally, I would like to acknowledge the BATWAL project (Convention 1318146, PE Plan Marshall 2.vert) financed by the Walloon region (Belgium) and the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA), since these were the funding sources that allowed me to perform this work.

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List of Figures

1.1 Air/airport incidents involving batteries carried as cargo or baggage organized according to a) the number of incidents every five years and b) the battery state of use (adapted from [5]). . . 3 2.1 Schematic representation of a cross section of a lithium-ion battery cell

subject to a charging process, highlighting the different battery com-ponents (adapted from [9]). Ions of lithium and hexafluorophosphate, and electrons are denoted by Li+, PF₆− and e−, respectively, while negative electrode, separator, positive electrode and battery cell thick-nesses are denoted by L−, Ls, L+and L, respectively. L−srepresents the negative electrode and separator thickness combined. . . 8 2.2 Schematic representation of the charging process of a lithium-ion

bat-tery cell. At the top: the batbat-tery cell domains (negative electrode (₋), separator (s) and positive electrode (+)), in the middle: the electrolyte phase (e), at the bottom: a particle solid phase (s). The indicated variables are: solid and electrolyte phase concentration (csand ce,

re-spectively), solid and electrolyte phase electric potential (φs and φe,

respectively), pore-wall molar flux (jn), electrolyte and solid-phase

diffusion coefficients (Deand Ds, respectively) and spherical particle

radius (Rs). . . 14

2.3 Schematic representation of main and side reactions, namely solvent re-duction reaction and lithium plating. Li+, Li(s)and e−denote lithium-ions, solid lithium and electrons, respectively, S and P are electrolyte solvent and side reaction product, respectively, and Rs and δf are

spherical particle radius and film thickness, respectively. . . 15 4.1 Equivalent-hydraulic model consisting of n tanks, where u, qi, gi,i+1

and βi, i=1, . . . , n, are respectively the input current, tank level state,

valve coefficient and tank cross-section area for a given number of tanks n. . . 56 4.2 Equivalent-hydraulic model representing the spherical particle with

n sections, where u and gi,i+1represent the input current and valve

coefficient, respectively. . . 57 4.3 Bode plots of the irrational transfer function Eq. (4.5) in black, and

from the 2nd to the 3rd-order Padé approximations of Eq. (4.17) and Table 4.2 in red, green and blue respectively. . . 59 4.4 Present situation of a BMS (purple region) together with the ideal

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4.5 Galvanostatic charge of a graphite|LCO battery cell at 5C. The left y-axis portrays the different negative electrode potential contributions to the battery voltage, as well as the side reaction potentials. The right y-axis shows the battery voltage. . . 65 4.6 Dependence of side reaction current density due to SEI growth on

surface overpotential. . . 70 5.1 Block diagram of the monitoring system for a battery cell. . . 73 5.2 Block diagram of the identification scheme. . . 74 5.3 a) Series of galvanostatic charge/discharge current profile of C/5, C/2

and 1C consecutive cycles; b) simulated voltage (noise free as solid black curve) and predicted voltage (dashed blue curve). . . 83 5.4 Stochastic stability simulation study considering three cases, namely:

1) small initial error and small measurement noise (solid red curve), 2) small initial error and large measurement noise (dashed green curve), and 3) large initial error and small measurement noise (dot-ted blue curve). The plot(dot-ted signals are a) series of galvanostatic charge/discharge current profile of C/5, C/2 and 1C consecutive cy-cles, and associated voltage (noise free simulation); b) simulated (solid black curve) and estimated CSC; c) estimation error of CSC. . . 84 5.5 Hola10 . . . 85 5.6 State estimation study: a) simulated (solid black curve) and predicted

CSC (dashed blue curve); b) CSC estimation error. . . 86 5.7 State estimation study: a) fixed (solid black curve) and predicted gs

(dashed blue curve); b) solid and dashed black lines respectively depict the lower and upper bound of the contact-resistance distribution Rr

while dashed blue curve is the predicted R_f. . . 87 5.8 a) Series of galvanostatic charge/discharge current profile of C/5, C/2

and 1C consecutive cycles carried out at 25◦C between 2.0 and 4.2 V; b) experimental (solid black curve) and predicted voltage (dashed blue curve). . . 90 5.9 State estimation study: a) Coulomb-counted (solid black curve) and

estimated SOC (dashed blue curve); b) estimated CSC. . . 90 5.10 State estimation study: a) estimated gs; b) experimentally determined

contact resistance mean ¯Rr(solid black curve) and estimated Rf (dashed

blue curve). . . 91 5.11 Block diagram of the fault detection and isolation scheme. . . 93 5.12 Nonlinear (in red), linearized (in green) and unscented transformed

(blue) mean (symbol) and covariance (curve) of normally distributed random variables for a) case 1, b) case 2 and c) case 3 in Table 5.8. The gray dots correspond to the sampled probability distribution of the random variable. . . 96 5.13 Primary reaction current state ( ˆz) estimated with the dual UKF in the

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5.14 State estimation study: a) solid-phase diffusion states; b) electrolyte-phase diffusion states; c) thermal states. Solid black and dashed red, green and blue (overlapping) curves represent the scenarios of fault-free, internal faults, voltage sensor fault and surface temperature sen-sor fault, respectively. . . 103 5.15 State estimation study: a) capacity fade state; b) normalized aging

parameters, namely rate capability and power fade on the left and right y-axis, respectively. Solid black and dashed red, green and blue curves represent the scenarios of fault-free, internal faults, voltage sensor fault and surface temperature sensor fault, respectively. . . 104 5.16 Fault detection study considering the voltage sensor: a) generated

residuals and b) associated GLR decision function. Solid red, dashed green and dotted blue curves represent the scenarios of internal faults, voltage sensor fault and surface temperature sensor fault, respectively. 105 5.17 Fault detection study considering the surface temperature: a)

gener-ated residuals and b) associgener-ated GLR decision function. Solid red, dashed green and dotted blue curves represent the scenarios of inter-nal faults, voltage sensor fault and surface temperature sensor fault, respectively. . . 105 6.1 Block diagram of the control system for a battery cell. . . 107 6.2 Block diagram of the optimal constrained control scheme. . . 108 6.3 Electrochemical constraints mapping, where a) nonlinear nonconvex

operating region delimited by constraint boundaries; admissible area for b) lower and c) upper bounds of constraint Eqs. (4.49a),(4.49b); union of linear approximations of constraint Eqs. (2.32c)-(2.32e), i.e. d) lines 5,1 and 5,2 and e) lines 6,1 and 6,2; and f) resulting admissible (green) and unsafe (red) regions resulting from the intersection of all the approximated constraints. . . 110 6.4 DFN model critical conditions for η+_srat the positive electrode/separator

interface (electrode dependent), η_sr− = 0.0 V at the negative elec-trode/separator interface and ce = 1.0 mol·m−3 at the current

col-lector/negative electrode interface (common for both batteries). The green, orange and red regions represent the admissible, safety margin and unsafe operating regions, respectively. The solid red and cyan curves represent the DFN simulator states and the EKF state estimates, respectively. . . 116 6.5 Current and voltage profiles from the DFN simulator using CCCV (as

dashed curves) and the proposed RG method (as solid curves) for LCO (in magenta) and LMO (in blue). . . 117 6.6 Profiles of the constrained variables from the DFN simulator using

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7.1 a) 2C CC applied current at 25◦C, and associated response of the battery in terms of b) voltage and c) surface temperature. Different system models are also shown in the voltage plots, namely Vexpis the

experimental voltage; V_mod,0 is the battery OCV; V_mod,1 is the surface overpotential; Vmod,2 is the electrolyte potential drop; and Vmod,3 is

an ohmic resistance potential drop. The temperature plots show the experimental surface temperature Tbs,expand the modeled core Tbc,mod

and surface temperatures Tbs,mod. . . 122

7.2 Flow chart of the cycling aging (CAT) and reference performance (RPT) tests. . . 126 7.3 a) Current and b) voltage response of a battery subject to C/2 CCCV

charging profile at 25◦C. . . 128 7.4 Schematic representation of the experimental layout needed to

im-plement the charging strategy based on the proposed RG with OR constraints scheme. . . 130 7.5 Simulink diagram for the implementation of the proposed RG with

OR constraints charging strategy. . . 132 7.6 Performance comparison between two commercial charging strategies,

namely 1C CCCV (in green) and 2C CCCV (in blue), in terms of a) retained capacity, b) resistance increase and c) charge time. Both CATs (curves) and RPTs (symbols) are shown. Dashed and solid curves respectively represent variables for the CC and the CCCV stage of a CCCV charging strategy. Symbols and # represent charge (chg) and discharge (dchg) variables, respectively. . . 133 7.7 Measurements gathered from the PEC tester for the first cycle block of

CAT, namely a) current, b) voltage (Vexp) and c) surface temperature

(Tbs,exp). Superimposed to these signals are the voltage (Vekf) and both

surface (Tbs,ekf) and core temperature (Tbc,ekf) predicted by the EKF. . . 136

7.8 EKF estimated internal state of the battery cell during the first cycle block, namely a) solid (SOC and CSC) and b) electrolyte-phase (Ceb

and Ces) diffusion states, and c) film resistance. The figure inset is a

close-up of the solid-phase diffusion state at the last cycle. . . 137 7.9 Signals generated by the RG during the first cycle block, namely a)

desired reference r and applied reference v, together with real SOC, and b) gain κ. . . 138 7.10 I-CSC plane portraying the trajectories followed by the battery

sys-tem state under the charging protocol generated from the RG during the first cycle block. The red-yellow color gradient reflects the cycle number from 1 to 11. . . 139 7.11 Performance comparison between 1C CCCV commercial (in green)

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7.12 Performance comparison between two commercial charging strategies, namely 1C CCCV (in green) and 2C CCCV (in blue), and the proposed RG (in red) charging strategy in terms of a) retained capacity, b) re-sistance increase and c) charge time. Both CATs (curves) and RPTs (symbols) are shown. Symbols and # represent charge (chg) and discharge (dchg) variables, respectively. . . 142 7.13 Performance comparison between two commercial charging strategies,

namely 1C CCCV (in green) and 2C CCCV (in blue), and the proposed RG (in red) charging strategy in terms of a) charging speed and b) capacity evolution with charge/discharge throughput. Both CATs (curves) and RPTs (symbols) are shown. Symbols and # represent charge (chg) and discharge (dchg) variables, respectively. . . 143 7.14 a) Current, b) voltage and c) surface temperature profiles from the

1st cycle block of a battery cell subject to different charging strategies, namely 1C CCCV (dashed green curves), 2C CCCV (dotted blue curves) and proposed RG (solid red curves). The figure inset is a close-up of the three signals during the 2nd charge cycle. . . 143 7.15 a) Current, b) voltage and c) surface temperature profiles from the

5th cycle per cycle block of a battery cell subject to different charging strategies, namely 1C CCCV (dashed green-olive curves), 2C CCCV (dotted pink-blue curves) and proposed RG (solid red-yellow curves). The first-second color corresponds to first-last cycle block, the color gradient reflects the cycle blocks in-between and the arrows in plot c) highlight the evolution of the thermal profile with cycle blocks. The figure inset is a close-up of the charging currents for the last cycle blocks during 2C CCCV charge. . . 145 7.16 Common stress factors considered for lithium-ion batteries, namely a)

maximum (solid curves) and average (dashed curves) applied current, b) maximum (dashed curves, denoted as "max") and end of charge voltage (solid curves, denoted as "EOC") and c) maximum surface temperature. Green, blue and red curves correspond to 1C CCCV, 2C CCCV and proposed RG, respectively. . . 146 7.17 Capacity fade correlation plot obtained by pairing the retained capacity

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List of Tables

2.1 Main DFN model equations. The nomenclature for this table is

pro-vided in the List of Symbols. . . 13

2.2 Two-states thermal model equations. The nomenclature for this table is provided in the List of Symbols. . . 14

2.3 Aging model equations for a generic side reaction sr, where sr = {SEI, l p}for SEI growth and lithium plating, respectively [11, 33, 34, 12] . The nomenclature for this table is provided in the List of Symbols. 16 3.1 The Kalman filter algorithm†. . . 30

3.2 The extended Kalman filter algorithm†. . . 32

3.3 The unscented Kalman filter algorithm†. . . 35

3.4 The standard instrumental variable algorithm†. . . 40

3.5 The simplified refined instrumental variable method for continuous-time systems algorithm, where s denotes the Laplace variable†. . . 43

4.1 Electrolyte-phase diffusion partial differential equations. . . 54

4.2 State Ajand input Bjmatrices for the diffusion equation of different orders, where j∈ {s, e}for solid and electrolyte phases, respectively. . 58

4.3 Functions associated to the material balance and output equations. . . 60

5.1 Extended Kalman filter with state constraints†. . . 77

5.2 The simplified refined instrumental variable method for continuous-time systems for the parameter identification of the transfer function Eq. (5.19) given the measurements Eq. (5.20)†. . . 80

5.3 Initial estimates and measurement noise variance for the stochastic stability study. . . 83

5.4 Results obtained using the SRIVC method for the estimation of Dsand Rs, and the LS method for the estimation of Rf on each cycle block of simulated data. . . 87

5.5 Fit results obtained from comparing the EKF estimation with the Padé approximation model using the true (θ) and estimated ( ˆθphys) parameters. 88 5.6 Results obtained using the LS method for the estimation of Dsand Rs on each cycle block of simulated data. . . 88

5.7 Results obtained using the SRIVC method for the estimation of Dsand Rs, and the LS method for the estimation of Rf on each cycle block of experimental data. . . 92

5.8 Mean and standard deviation values of the random variables used to simulate model Eqs. (5.40),(5.41). . . 95

5.9 The dual unscented Kalman filter algorithm for the considered NLDAE system [156, 126]†. . . 100

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6.1 Values for the DFN model constraints Eqs. (2.32) associated to degra-dation mechanisms. . . 115 6.2 Values for the constraints Eq. (6.9). . . 115 7.1 Instrumentation accuracy of the battery tester, namely current, voltage

and temperature sensor, as well as data sampling time. . . 128 7.2 Fitting parameters of the linear function Eq. (7.20) describing the

retained capacity for a battery subject to 1C and 2C CCCV charging strategies. The fitted data is the RPT points of Fig. 7.6(a), and a set of parameters is obtained for each C-rate and cycle stage. . . 134 7.3 Fitting parameters of the linear function Eq. (7.21) describing the

resistance increase for a battery subject to 1C and 2C CCCV charging strategies. The fitted data is the RPT points of Fig. 7.6(b), and a set of parameters is obtained for each C-rate and charge/discharge mode. . . 135 7.4 Fitting parameters of the linear function Eqs. (7.20) and (7.21)

describ-ing the retained capacity and resistance increase for a battery subject to the proposed RG charging strategy. The fitted data is the RPT points of Fig. 7.11(a), and a set of parameters is obtained for each cycle stage and charge/discharge mode. . . 140 8.1 Summary of the different contributions proposed throughout this thesis.151 C.1 List of parameters for the LFP half battery cell [148] used in Section 5.2

to set up the DFN-based simulator. . . 159 C.2 List of parameters used for setting up the DFN-based simulator used in

Section 5.4 and Chapter 6 for a graphite|LCO [157] and a graphite|LMO [22, 157, 159] battery cell. . . 160 C.3 List of parameters used for setting up the DFN-based simulator used

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List of Abbreviations

BMS Battery-Management System

CC Constant Current

CV Constant Voltage

CAT Cycling Aging Test

CSC Critical Surface Concentration

DAE Differential Algebraic Equation

DFN Doyle-Fuller-Newman

ECM Equivalent-Circuit Model

EKF Extended Kalman Filter

EChM ElectroChemical Model

FDI Fault Detection and Isolation

GLR Generalized Likelihood Ratio

IV Instrumental Variable KF Kalman Filter LS Least Squares LCO Lithium-Cobalt-Oxide LFP Lithium-Iron-Phosphate LMO Lithium-Manganese-Oxide

LQR Linear Quadratic Regulator

LTO Lithium-Titanium-Oxide

LRPT Long Reference Performance Test

MPC Model Predictive Control

NCA Nickel-Cobalt-Aluminum

NMC Nickel-Manganese-Cobalt

NLDAE NonLinear Differential Algebraic Equation

OCP Open Circuit Potential

OCV Open Circuit Voltage

ODE Ordinary Differential Equation

PDE Partial Differential Equation

RG Reference Governor

RPT Reference Performance Test

SEI Solid-Electrolyte Interphase

SIV Standard Instrumental Variable

SOC State-Of-Charge

SOH State-Of-Health

SPM Single Particle Model

SRG Scalar Reference Governor

SVF State Variable Filter

SRPT Short Reference Performance Test

SRIVC Simplified Refined IV method for Continuous-time

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Physical Constants

Universal gas Rg=8.31 J·mol−1·K−1

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List of Symbols

Electrochemical modeling

A cross-sectional area of the cell m2

as specific interfacial area m−1

c concentration of lithium mol·m−3

Cp specific heat capacity J·g−1·K−1

CSC critical-surface-concentration

-D diffusion coefficient m2·s−1

E activation energy J_·mol−1

f_c/a activity coefficient

-g inverse of the diffusion time constant s−1

hc heat transfer coefficient W·m−2·K−1

i0 exchange current density A·m−2

I applied current A·m−2

j pore-wall molar flux mol_·m−2_·s−1

k reaction rate constant A·m2.5·mol−1.5

kc thermal conductivity W·m−1·K−1

L electrode/separator thickness m

M molecular weight g·mol−1

nLi number of lithium moles mol

Q_b battery capacity Ah

Qe electrode capacity Ah

Qloss battery capacity loss Ah

r radial coordinate m R particle radius m Rb battery resistance Ω R_f film resistance Ω_·m2 SOC state-of-charge -t time s t0_c transference number

-Tamb ambient temperature K

T_b battery temperature K

U open-circuit potential V

V voltage V

x coordinate across the cell thickness m

y stoichiometry of lithium

-∇ cutoff frequency Hz

System monitoring and control â estimated value of a ã error a₋ â

¯a mean of a

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a+ projected value of a k discrete-time variable t continuous-time variable t_k sampled data time variable t0 initial time tf final time Ts sampling period r desired reference u input vector v applied reference

vn measurement noise vector

wn process noise vector

x dynamic state vector

y output vector

z algebraic state vector

A state matrix

B input matrix

C output matrix

F state function Jacobian H output function Jacobian Id d by d identity matrix

K Kalman gain

Ki LQR gain for integrator state

Kc LQR gain= [Kc, Ki]

Kx LQR gain for dynamic state

P estimation error covariance matrix

Pc solution to the control algebraic Riccati equation

Q process noise variance matrix Qc tracking error weighting matrix

R measurement noise variance matrix Rc control effort weighting matrix

f(·) nonlinear function characterizing the state dynamics

h(_·) nonlinear function in the output

equation or the algebraic state equation G(s) transfer function

cc total number of constraints

nc number of OR constraints

f fault magnitude

gGLR generalized likelihood ratio decision function

¯h threshold

S(·) cumulative sum Greek

α0 apparent transfer coefficient

-ε active material volume fraction

-δ thickness m

δt time delay s

e Bruggeman’s exponent

-ζ instrumental vector

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φ electric potential V ϕ regressor vector

κe ionic conductivity S·m−1

κ reference governor gain

ν residual

ρ density g·m−3

σ electronic conductivity S·m−1

τ diffusion time constant s

τf,0 time instant of fault injection s

Υ measurement

Subscript

b bulk

c core

cl closed-loop

coef polynomial coefficient e electrolyte phase eff effective variable

f film

f filtered signal

l linearized

l p lithium plating max theoretical maximum

n main reaction

nl nonlinear

p side reaction product phys physical ref reference (_·)s surface s solid phase sr side reaction st solvent

SEI solid-electrolyte interphase

t total T thermal UT unscented transformed θ parameter Superscript ac active constraint d discrete-time s separator domain

− negative electrode domain

+ positive electrode domain Math notation

|| · || 2-norm

1n×1 column vector of ones with n components

0n×1 column vector of zeros with n components

a mean or upper bound

a lower bound

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δx error on the measurement of x

∆χ± _{difference between χ}+_{and χ}₋_variables diag(_·) diagonal elements of the matrix entry

V (_·) AND operator W (·) OR operator E(·) expectation operator H(·) hypothesis L(·), Pθ probability law

N (µ, σ2) normal distribution with mean µ and variance σ2

X sigma-points for x

O∞ maximal output admissible set ˜

O_∞ inner approximation of O_∞

Yk−1 data sequence up to and including k

N natural numbers R real numbers R2 _{coefficient of determination} s Laplace variable σ standard deviation σ2 variance

pθ probability density function of θ

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Chapter 1

Introduction

This document shows the improved performance and increased lifetime that can be extracted from lithium-ion batteries through their advanced management. More precisely, electrochemical models and degradation constraints are exploited to develop a battery monitoring and control system that is health-conscious and does not cut down on performance. These two aspects are highly relevant specially for large scale applications, such as electric vehicles and renewable energy storage systems. An improved operation of the battery has the potential to greatly reduce both the price of the system and the time of recharge, and increase both the battery lifespan and safety. Having a product with all these assets is very appealing from the point of view of the investor or consumer.

The proposed battery management system relies on physical models and control theoretic tools. On the one hand, various electrochemical models differing in com-plexity are studied in detail, together with model constraints that trigger degradation processes if violated. On the other hand, electrochemistry-based state observers, fault diagnostic systems and constrained controllers are developed and tested notably for the fast and safe charging of the battery. These solutions are not limited to any particular lithium-ion battery chemistry, and pieces of them can be used for problems in other energy storage related research topics like fuel cells.

This work pinpoints the main sources of battery degradation from the electro-chemical perspective, and proposes ways to handle them during battery regular operation. It turns out that some degradation mechanisms cannot be avoided for a reasonable battery use, and they reduce battery capacity but do not imply a safety hazard. Other mechanisms however are more detrimental to both battery life and safety, but they can be mitigated through health-conscious strategies. Novel methods for battery monitoring and control are presented in this dissertation, which are for the first time (to the best of the author’s knowledge) validated with a battery-in-the-loop experimental setup. Indeed, the proposed approach is able to charge faster and safer lithium-ion batteries when compared to commercially available charging strategies as CCCV protocols.

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1.1 Motivation

Among the main concerns regarding battery systems are their price, safety, reliability and performance, not necessarily in this order of priorities. In this context, the work proposed here aims at providing solutions to these issues through the development of advanced battery management systems (BMSs). A lithium-ion battery controlled by an advanced BMS might be designed with less conservatism for the same performance as one operated by a traditional BMS. This aspect has the potential to reduce both the battery size and price, which would have a huge impact in large scale applications like electric vehicles. In fact, automotive applications have become the driving force to impulse the outbreak of lithium-ion batteries in other sectors, such as the grid and renewable energy storage. With battery cell prices ranging from 200 $·kWh−1to 500 $·kWh−1[1], the price of the battery can amount from 15% to 50% of the cost of an electric car1. A few dollar difference per kWh can have big repercussions for the car cost. In a market that is expected to boom from 3 to 125 million electric vehicles on the road worldwide by 2030 (according to the International Energy Agency), battery price may enable a thriving technology or doom it to fail.

Even though battery price or size are important factors when considering this energy storage technology, other concerns are by far more important such as safety and lifetime expectancy. In terms of safety, battery failures like short circuits can have serious consequences, as exemplified by the lithium-ion battery failures in certain Samsung mobile phones [2] or the blaze due to a battery in a Boeing 787 Dreamliner in 2013 at Heathrow Airport [3], just to name a few. One of the main agencies that have kept track of different battery incidents is the Federal Aviation Administration of the United States. A total of 225 air/airport incidents involving batteries carried as cargo or baggage have been recorded from 1991 to 2018 [4]. Fig. 1.1(a) depicts the number of incidents registered every five years starting from 1991, whereas Fig. 1.1(b) shows a pie chart with the incidents organized according to the battery state of use, namely new, in use by consumers or in equipment batteries. As the figure shows, there has been an increase in the amount of incidents related to batteries since 1991. Most of these incidents are associated to new batteries.

In terms of battery life span, batteries are expected to last as long as the application in which they are used, for instance 10-15 years (i.e. 20,000-30,000 cycles) of service life in automotive applications [6], and 20 years as autonomous power supply systems for solar panels [7]. Safer batteries with prolonged lifetime but without trading performance is what we look for with this work. This can be achieved via advanced monitoring and control tools.

1.2 Open Challenges

Although a rich body of literature has been already built on the topic of lithium-ion batteries development, monitoring and control, there are still some challenges that need to be faced, namely:

• The term "lithium-ion" does not denote a battery but a family of batteries that share the lithium chemistry. This issue complicates the modeling, monitoring and control problem, which nowadays counts with a myriad of solutions in the

1_{This price share has been obtained by considering battery prices ranging from 200 $}_·_kWh−1_to

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a)

b)

FIGURE1.1: Air/airport incidents involving batteries carried as cargo or baggage organized according to a) the number of incidents every

five years and b) the battery state of use (adapted from [5]).

literature. An unified framework to develop battery management systems built upon the most performant methods is still a pending task.

• Models that are highly descriptive of the processes taking place inside a bat-tery are based on electrochemistry. These models are infinite-dimensional and nonlinear, and such sophistication prevents their exploitation for online battery monitoring and control. Reduced-order electrochemical models able to keep rel-evant battery dynamics exist, but they are typically limited to specific operating ranges.

• Online monitoring of internal states and degradation is difficult during battery regular operation given the limited number of measured quantities and the effect of measurement noise. Moreover, estimation schemes completely rely on the accuracy of the taken measurements and overlook possible instrumentation malfunctions. Diagnostic systems able to provide fault detection and robust and unbiased estimation are missing.

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This aspect still needs to be recognized, and solutions along these lines should be provided.

1.3 Contributions

In this thesis, we aim at designing and validating an advanced monitoring and control system for lithium-ion battery management that improves battery perfor-mance and lifetime. The derived control schemes are tested with different battery chemistries in order to highlight the generality of the proposed framework. To achieve this goal, three main categories of contributions can be distinguished, which are shown in the following and also gave rise to the list of publications reported in Appendix A.

• Electrochemical modeling of lithium-ion batteries: A full-order electrochemi-cal model based on nonlinear partial differential algebraic equations is modified to account for degradation processes. This high-order model is then reduced by resorting to physically motivated approximations to come up with simpler models that are useful for monitoring tasks. Different aging mechanisms are considered and categorized according to their appearance during battery regu-lar operation. Each mechanism is treated in a particuregu-lar way. Electrochemically meaningful constraints associated to battery degradation are derived, so that their compliance improves the performance and extends the life of the battery. Even though different reduced-order electrochemical models have been pro-posed in the literature, the original one used here and later extended has never been associated to degradation mechanisms. Moreover, the clear distinction between battery aging tracking and degradation avoidance has been drawn and exploited here for the first time.

• Battery diagnosis system: State observers based on electrochemical models are designed, which account for parametric uncertainty, operating point changes, algebraic constraints arising from battery degradation and physical constraints. Although the state observers used are standard, their novelty with respect to the ones in the literature lies in i) the electrochemical models at their core, ii) the way of combining different design strategies to address the battery estimation problem at hand, like equality and inequality state constraints, and iii) the developed physics-based tuning procedure. Battery state-of-health indicators are deduced from the physics, and their estimation is tackled. Tools coming from the parameter identification community are exploited in order to properly handle the initialization and noise susceptibility (Refs. A.1-A.3) of more traditional parameter estimation approaches proposed by other authors. Sensor fault detection and isolation is also carried out in parallel to state-of-health estimation, which has been mostly overlooked by the battery research community so far (Ref. A.4).

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a faster and safer way than commercial charging protocols based on CCCV protocols. This latter claim was verified via long-term battery cycling tests. This is the first time that reduced-order electrochemical models, state observers and constrained control schemes as the ones proposed in this dissertation have been proven successful experimentally for the advanced management of the charging process of lithium-ion batteries.

• Besides the work reported in this thesis, I also contributed to other research projects, see for instance Refs. A7-A11.

1.4 Thesis Structure

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Chapter 2

State of the Art

The research concerning lithium-ion batteries is multidisciplinary, from battery development, passing by the ancillary electronics surrounding its operation (actua-tors, instrumentation), up to the software with suitable algorithms to monitor and control the working battery. At least three main bodies of literature can be distin-guished in the field of batteries, each one of them associated to a major field in science. The first body comes from electrochemistry, and it focuses on the material science aspect of developing batteries. The resulting batteries are further tested to provide insight into the processes occurring inside them, i.e. how a given battery works and what limits its performance. The second body comes from electrical and electronic engineering, and it is related with the battery integration into a given power network, either a battery cell within a battery pack or a pack within an electrical network. The last body comes from automation and control engineering, and it is framed in the context of battery-management systems (BMSs). There is no consensus of the final definition and functions of a BMS, but a wide view adopted in [8] is also considered here, i.e. a BMS is any system that manages the battery. Among its tasks, a BMS has to protect the battery cells from being damaged, to guarantee their safety and prolong their service life as long as possible and to ensure that they fulfill the application requirements. The BMS involves then the development of software in charge of monitoring and controlling the battery during regular operation. This dissertation is centered around the first and third battery main topics, namely it exploits the knowl-edge from the electrochemical community to develop highly performant supervisory control systems.

In the following, the state of the art in the fundamental research of batteries and their supervisory control is covered. The former topic involves a description of the battery operation and of the source responsible of limiting its performance in the long run, i.e. battery aging. The latter topic includes battery modeling, state/parameter estimation, fault detection and isolation and optimal control, subtopics that are discussed subsequently.

2.1 Battery Operation

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nickel-cobalt-aluminum (NCA) or nickel-manganese-cobalt (NMC), and blends of those. Besides these active materials, electrodes also incorporate a binder and a conductive filler to improve electrode particles cohesion and electric conductivity, respectively. The separator can be made of polyolefin materials such as polyethy-lene and polypropypolyethy-lene. It is placed in-between the two electrodes to isolate them electronically, although it is an ionic conductor. The electrolyte can be liquid or gel, and it is a solution of lithium salts (e.g. lithium hexafluorophosphate, LiPF6) and

organic solvents (e.g. carbonates such as ethylene and diethyl carbonate). Current collectors are placed at the negative and positive terminals of the battery cell. Copper and aluminum are the most commonly used materials for negative and positive electrodes, respectively. They both are electronic conductors.

During charging, a positive overpotential is applied to the battery cell, which promotes the deintercalation (oxidation) of lithium from the positive electrode. This process corresponds to the forward reaction in Eq. (2.1) for a generic graphite|LMeO battery, where Me stands for the metal of a given chemistry [9].

C6+xLi++xe− LixC6

LiMeOz Li1−yMeOz+yLi++ye−. (2.1)

The resulting lithium-ions (Li+) constitute an ionic current that travels throughout the electrolyte. It leaves the positive electrode, crosses the separator and reaches the negative electrode, where the intercalation (reduction) of lithium takes place. Simul-taneously, an electric current leaves the system from the positive electrode, performs electrical work and comes back to the system through the negative electrode. This process is depicted Fig. 2.1. The opposite mechanism is the discharge process, which corresponds to the backward reaction in Eq. (2.1). The stored energy is a function of the potential difference between both electrodes. Both oxidation/reduction reactions constitute the desired intercalation reaction in a lithium-ion battery cell.

FIGURE2.1: Schematic representation of a cross section of a lithium-ion battery cell subject to a charging process, highlighting the dif-ferent battery components (adapted from [9]). Ions of lithium and hexafluorophosphate, and electrons are denoted by Li+, PF₆−and e−, respectively, while negative electrode, separator, positive electrode and battery cell thicknesses are denoted by L−, Ls, L+and L, respec-tively. L−srepresents the negative electrode and separator thickness

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2.2 Battery Aging

Besides the main intercalation reaction, lithium-ion batteries are prone to side reactions occurring simultaneously. The latter reactions consume available lithium, which leads to capacity/power fade and can even compromise the battery safe operation [9, 10]. Battery degradation might have different origins, which could be correlated and are dependent on the specific chemistry [9]. The most relevant sources of degradation that have been identified for a variety of electrode chemistries include solid-electrolyte interphase (SEI) formation and growth [11], and lithium plating [12].

The SEI is a passive film that is formed on the surface of the negative electrode during the first charging processes. Such a film is the product of reduction side reactions involving lithiated carbon, the electrolyte and the solvent. At the beginning, the SEI protects the electrode from further reacting with the electrolyte and slows down the battery degradation rate. However, in the long run the SEI limits battery performance through capacity and power fade. The reduction of solvent ethylene carbonate S occurring at the graphite electrode can be modeled as [11]

S+2Li++2e−_→P (2.2)

where P is the obtained product.

Another side reaction that consumes available lithium is the so-called lithium plating, and this one is by far more dangerous than SEI growth. Lithium plating occurs when the potential of the graphite negative electrode becomes negative, and it consists of lithium deposition onto the electrode surface instead of its intercalation. This deposition of metal lithium can lead to dendrites formation, which can pierce the separator and cause a short circuit. It turns out that lithium plating occurs at later stages of battery life, and it is responsible for the nonlinear aging behavior evidenced at such stages [12]. This reaction can be written as [13]

Li++e− _→Li(s). (2.3)

where Li(s)is metal lithium.

Apart from side reactions resulting in SEI growth and lithium plating, some other relevant degradation mechanisms include electrolyte decomposition, SEI breakdown, gas generation, overcharge, overdischarge, self-discharge, loss of active material, mi-gration of soluble species, particle fracture, mechanical stress and structural changes [9, 14, 15, 16, 17].

Battery degradation can occur either when batteries are in use or even when they are stored. The former operation condition is known as cycling aging while the latter is calendar aging. Empirical models for calendar aging assume that dominant degradations like impedance (denoted Rf) increase due to SEI growth and lithium

capacity loss Q_losshave a power of time relationship such as

Rf ∝ a1tz, Qloss∝ b1tz (2.4)

where a1 and b1 are constants, z = 1/2 for diffusion-controlled and z = 1 for a

kinetic controlled aging processes, and it takes values of z = [0.3, 1] for mixed mechanisms [18]. Empirical models for cycling aging assume that degradation rates are proportional to cycle number N such as

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where a2and b2are constants, in general p=0 for Rf whereas it takes values within

the interval[0, 3]for Qlossaccording to the acceleration rate of the degradation process

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2.3 Battery Modeling

In order to perform battery monitoring and control, accurate models are required. The objectives of battery modeling are twofold. In one hand, battery simulators based on sophisticated models allow the numerical validation of control strategies, which on the other hand are built upon simpler models. Several lithium-ion battery cell models have been proposed in the literature, from data-driven [19] to equivalent-circuits [20] and electrochemical models. The first type of model has the drawback of requiring a large amount of data for training. Equivalent-circuit models (ECM) rely on the analogy between battery systems and electrical systems. However, the former systems involve lithium diffusion, migration, intercalation and reaction processes, besides the electrical ones, which cannot be modeled by the latter type of systems. Therefore, ECM variables suffer from lack of interpretability, preventing their link to e.g. the physics of battery degradation. Physically meaningless models cannot be exploited for developing high-fidelity battery simulators rooted in electrochemical principles. These inconveniences are not exhibited by the physics-based electrochemical models (EChMs). Furthermore, in contrast to other models, EChMs can be leveraged to perform battery fault diagnosis and constrained control that is less restrictive.

In the following, we only focus on electrochemical type of models in order to limit the scope of this work. Therefore, we deliberately leave aside a rich body of literature dealing with other modeling efforts such as ECMs. Interested readers are referred to [20] for a comprehensive review in ECMs. Regarding electrochemical models, they have been less exploited historically, although recent years have witnessed a boom in their development and use.

The Doyle-Fuller-Newman Model

Among the different EChMs, the Doyle-Fuller-Newman (DFN) model [21, 22, 23] is the most widely used one. The DFN model mathematically describes the electro-chemical processes occurring inside a lithium-ion battery cell, which are detailed in Section 2.1. It considers the negative electrode(−), the separator(s)and the positive electrode(+)as three domains, with two phases, namely the porous solid phase(s)

and the electrolyte solution phase(e)1_{. The solid phase of the system is assumed to}

be made of spherical particles. While the solid phase is only present in the electrode domains, the solution phase covers all three domains. Fig. 2.2 depicts a schematic representation of the charge process taking place within a lithium-ion battery cell. The whole cell with the three domains and two phases is shown at the top, the elec-trolyte phase is in the middle, and a spherical particle taken out from each electrode solid-phase is depicted at the bottom of the figure. The main partial differential and algebraic equations describing the electrochemical system dynamics are shown in Table 2.12. Each of these five equations has an associated dependent variable, which is also represented in Fig. 2.2 using the same notation as the table. These variables are: the solid and electrolyte phase concentration (csand ce, respectively), the solid and

electrolyte phase electric potential (φsand φe, respectively) and the pore-wall molar

flux (jn). All these variables depend on the space (r- and x-axis along the radius of the

electrodes spherical particles and the thickness of the battery cell, respectively) and the time (t). This model is a multi-particle model, in which a solid-phase spherical

1_{The symbols in parentheses are used as super-script and sub-script to denote the domain and}

phase of a model state variable, respectively.

2_{Continuous-time with time variable t and the time derivative as}dχ

dt(·, t)is adopted in this section

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particle can be located at each x position in the electrode domains. Within these spherical particles, diffusion takes place only in the r direction. This fact explains the other name with which the DFN model is known, i.e. pseudo-2D model.

Remark 2.3.1. Even though the DFN model is the most widely used battery model based

on electrochemistry, it has some limitations. Firstly, it is a pseudo-2D model where lithium-ions diffuse in 1D along the battery cell thickness, and it assumes homogeneous diffusion throughout the spherical particle radius. In large format batteries, a 1D description of the cell might not be sufficient to accurately capture its thermo-electrochemical behaviour. Thus, the DFN model has been extended to pseudo-3D [24] and 3D [25] models. Secondly, the original DFN model neglects some battery dynamics, such as thermal and aging dynamics, but it has been proven flexible to include them as additional phenomena, see for instance [26] for temperature evolution, [11] for solid-electrolyte interphase growth or [17] for intercalation-induced stress in active particles. More recently, the DFN model accuracy has been put in question for high currents and temperatures [27]. In spite of these restrictions, the DFN model still constitutes the virtual battery benchmark for excellence and therefore it is exploited in this dissertation for battery cell simulation. The considered DFN model is extended with thermal dynamics and some degradation mechanisms as explained below.

The charge process in Fig. 2.2 shows, at the bottom plot, lithium diffusion from the bulk to the surface of the positive electrode spherical particle, where the main deintercalation reaction occurs and produces a pore-wall molar flux. The lithium then passes to the electrolyte phase (see middle plot), where it is in ionic form and travels from the positive electrode, through the separator up to the negative electrode. Once in the negative electrode, a pore-wall molar flux is produced while the main intercalation reaction takes place, followed by lithium diffusion from the surface to the bulk of the negative electrode spherical particle, as it can be seen again at the bottom plot. Meanwhile, the top plot portrays the electrons journey from the positive electrode to the negative electrode, generating electrical work outside the system.

While the original DFN model framework [21, 22] is the basis for battery cell modeling, other equations have been incorporated into this framework to capture more dynamics. For instance, thermal models derived from energy balance [28, 29, 30] have been coupled to the DFN model by replacing the constant temperature Tamb

by a given dynamic battery temperature state Tb. Although these thermal balances

consider a bulk temperature state with lumped thermal parameters, other works discriminate between the battery core (bc) and surface (bs) temperature3[31]. The main differential and algebraic equations describing the thermal system dynamics are shown in Table 2.2.

Other dynamics introduced in the DFN framework correspond to those of aging mechanisms, such as the ones described in Section 2.2. The equations behind side reactions (sr) have been formally derived from electrochemical principles. The main equations describing the aging induced by side reactions such as SEI growth (SEI) and lithium plating (l p)3are shown in Table 2.3. The cathodic Tafel approximation for the side reaction rate Eq. (2.22) is deduced by considering the Butler-Volmer kinetics (Eq. (2.14) considering α0 = 0.5) under the assumption of an irreversible solvent

reduction reaction [23]. Moreover, although some models only consider side reactions during battery charge claiming that they are more relevant during this operation mode, here we opted for allowing battery degradation during discharge as well [32]. This approach avoids the degradation discontinuity depending on the operation mode and lets the side reaction to take place as a function of its given overpotential.

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FIGURE2.2: Schematic representation of the charging process of a lithium-ion battery cell. At the top: the battery cell domains (negative electrode (−), separator (s) and positive electrode (+)), in the middle: the electrolyte phase (e), at the bottom: a particle solid phase (s). The indicated variables are: solid and electrolyte phase concentration (cs

and ce, respectively), solid and electrolyte phase electric potential (φs

and φe, respectively), pore-wall molar flux (jn), electrolyte and

solid-phase diffusion coefficients (De and Ds, respectively) and spherical

particle radius (Rs).

TABLE2.2: Two-states thermal model equations. The nomenclature for this table is provided in the List of Symbols.

Physical Process Equation

Core Temperature ρcCpcdTbc dt (t) =kc(Tbs(t)−Tbc(t))−I(t)V(t) − L Z 0 a±_s Fj±_n(x, t)∆T(¯c±_s , Tbc)dx (2.18) where ∆T(¯c±_s , Tbc) =∆U_b±(¯c±s (x, t))−Tbc(t)∆ ∂U_b± ∂Tc (¯c±_s(x, t)) Surface Temperature ρsCps dTbs dt (t) =kc(Tbc(t)−Tbs(t)) +hc(Tamb−Tbs(t)) (2.19) Arrhenius Equationa Φ(Tbc) =Φrefexp

E_Φ Rg 1 T_ref − 1 Tbc(t) (2.20) a_{Φ could be D} s, De, kn, κeor i0,sr

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schematically represented in Fig. 2.3(a) and (b), (c), respectively. Two distinct side reactions are portrayed in the figure, namely solvent reduction reaction of Eq. (2.2) in Fig. 2.3(b) and lithium plating of Eq. (2.3) in Fig. 2.3(c). The former side reaction involves lithium-ions Li+and the electrolyte solvent S, and it creates an insoluble product P (in dark gray in Fig. 2.3(b)) whose accumulation gives rise to the SEI film. The latter side reaction generates solid lithium Li(s)(in light gray in Fig. 2.3(c)). Both P and Li(s)are deposited at the spherical particle surfaces (dark orange arc in Fig. 2.3). It is worth noticing that the conductive nature of the SEI film conditions the degradation response of the battery, as it is depicted in the close-up inset of Fig. 2.3(b). If the film conducts well electricity (e−represents electrons) but poorly ions, lithium-ions and electrons likely undergo the main reaction at the surface of the film (upper reaction in Fig. 2.3(b) inset). Then, the film is said to be diffusion-controlled, and the lithium diffusion time is effectively increased from τs = R

2 s

Ds to τs=

(Rs+δf)2

Ds ,

where Ds, Rsand δf are respectively the solid-phase diffusion coefficient, spherical

particle radius and film thickness. The arrow associated to δf pointing outwards

indicates how the film gets thicker with the passage of time. Conversely, if ions get across easily and electrons have a harder time to go through the film, the main reaction occurs at the surface of the particle (lower reaction in Fig. 2.3(b) inset). The film is then said to be kinetic-controlled, and the battery voltage suffers an ohmic potential drop caused by the film resistance Rf. A mixed degradation mechanism

happens in-between these two extremes. Empirical models representing the battery response to these limiting degradation conditions in terms of time and cycle number were shown above in Eqs. (2.4) and (2.5).

FIGURE 2.3: Schematic representation of main and side reactions, namely solvent reduction reaction and lithium plating. Li+, Li(s)and e−denote lithium-ions, solid lithium and electrons, respectively, S and P are electrolyte solvent and side reaction product, respectively, and Rsand δf are spherical particle radius and film thickness, respectively.

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TABLE2.3: Aging model equations for a generic side reaction sr, where sr = {SEI, l p}for SEI growth and lithium plating, respectively [11, 33, 34, 12] . The nomenclature for this table is provided in the List of

Symbols.

Physical Process Equation

Current Balance jt(x, t) =jn(x, t) +

∑

i∈sr ji(x, t) (2.21) Side Reaction Rate jsr(x, t) =−asLi0,srexp − α0F RgTbc(t) ηsr(φs, φe, jt) (2.22) Side Reaction Overpotential ηsr(φs, φe, jt) =φs(x, t)−φe(x, t)−Usr−Rf(x, t)Fjt(x, t) (2.23) Conservation of Solvent ∂cst ∂t (x, r, t) =Dst ∂2cst ∂r2 (x, r, t)− dδ_f dt (x, t) ∂cst ∂r (x, r, t) (2.24)

with boundary conditions

−Dst ∂cst ∂r (x, r, t) r=Rs +dδf dt (x, t)cst(x, r, t) =jsr(x, t) cst(x, r, t)|r=Rs+δf=εfc b st(x, r, t)

Capacity Loss dQloss

dt (t) =− L Z 0 FasA 3600jsr(x, t)dx (2.25) Film Growth dδf dt (x, t) =− Mp ρp jsr(x, t) (2.26) Impedance Rise Rf(x, t) =Rf(0) + 1 κp δf(x, t) (2.27)

variations can be state dependent or take the form of empirical functions. Examples of the former variations are the side reaction-dependent electrolyte volume fraction [34] and solid-phase diffusion coefficient [35] given respectively by

dεe dt (t) = 1 2 Mp ρp 3εs Rs jsr(t), (2.28) dDs dt (t) =− n1Ds(0) εs(0) εs(0)−εs(t) εs(0) n₁−1 Mp ρp asjsr(t) (2.29)

where Mpand ρpare the side reaction product molar mass and density, respectively, εs and εe are active material volume fraction for the solid and electrolyte phases,

respectively, Dsand Rsare the solid-phase diffusion coefficient and particle radius,

respectively, asis the specific interfacial area and n1is an empirical factor representing

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[37] are given respectively by dεs dt(t) =C1(Tb)|Fjn(t)| +C2(Tb) q |Fjn(t)|, (2.30) Ds(t) =k1exp k2 N (2.31) where C1(Tb)and C2(Tb)are fitting parameters that depend on temperature, k1and

k2 are constants and N is the cycle number. These equations are just examples of

parametric variations linked to aging mechanisms.

Operating Constraints

Whether battery degradation comes from side reactions or parameter changes, the capacity/power fade to which the battery is subject to can compromise its safe operation [10, 9]. Interestingly enough, battery failure can be mitigated by avoiding certain operating regions where degradation is boosted. Such delimitation of the safe operating regions translates in constraints, which can be exploited by a controller to steer the battery state inwards the safe region. For the charging process, the constraints take the form

              

¯c−_s (x, t)/c−_s,max ≤rcs, c−ss(x, t)/c−s,max ≤rcs (2.32a)

¯c+_s (x, t)/c+_s,max ≥r_c_s, c+_ss(x, t)/c+_s,max ≥r_c_s (2.32b)

η−_sr(x, t) =φ_s−(x, t)−φe(x, t)−Usr−>0 (2.32c) η+_sr(x, t) =φ_s+(x, t)−φe(x, t)−Usr+<0 (2.32d)

ce(x, t)≥rce (2.32e)

where ¯cs and css are the average and surface lithium concentration, respectively,

whose normalization with respect to the maximum allowed lithium concentration cs,maxare lower and upper bounded by two constants rcs and rcs, respectively, that

depend on the electrode chemistry. The overpotential and the equilibrium potential of the side reactions are denoted by ηsr and Usr, respectively. Eqs. (2.32a) and (2.32b)

hinder the situation where Li is extracted/deposited beyond the maximum concentra-tion allowed by the electrode, which can cause phase transformaconcentra-tions, active material dissolution and oxygen loss in positive electrodes, and lithium dendrite formation in negative electrodes. Eqs. (2.32c) and (2.32d) prevent side reactions from taking place, which consume cyclable lithium and reduce the cell capacity (i.e. capacity fade) [38]. Such conditions occur first at the negative electrode/separator and the positive electrode/separator interfaces, respectively. Eq. (2.32e) precludes electrolyte depletion, which takes place first at the current collector/negative electrode.

Reduced-Order Models

Electrochemical Modeling, Supervision and Control of Lithium-Ion Batteries